Zee is, as usual in his book "QFT in a nutshell" not quite right.
First of all, current conservation is weaker than gauge invariance since it follows already from a global symmetry. E.g., take the Dirac-Yukawa Lagrangian
\mathcal{L}_{\text{Dirac}} = \overline{\psi}(\mathrm{i} \gamma^{\mu} \partial_{\mu} - m \psi + \frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi) - g \overline{\psi} \psi \phi.
Since this Lagrangian is invariant under the global transformation
\psi \rightarrow \exp(\mathrm{i} \alpha) \psi, \quad \overline{\psi} \rightarrow \exp(-\mathrm{i} \alpha) \overline{\psi}, \quad \phi \rightarrow \phi
with \alpha=\text{const}, the usual current j^{\mu} = \overline{\psi} \gamma^{\mu} \psi is conserved, but for sure this Lagrangian is not invariant under local gauge transformations, i.e., you cannot make \alpha a function of the space-time variables x since the term with the derivative of the Dirac field leads to an additional term which needs to be compensated by the introduction of a gauge field, which is a vector field, with help of which you can define a covariant derivative via \partial_{\mu} + \mathrm{i} q A_{\mu}.
In the most simple form, you keep the vector field massless, and this is natural in the sense that a massless vector field with a finite number of spin-like degrees of freedom (for usual massless vector particles these are the two polarization states, e.g., given by the states of helicity 1 and -1), from Poincare invariance the massless vector field must necessarily be introduced as a gauge field.
Local gauge invariance has important consequences for the Green's and the 1PI vertex functions, the already mentioned Ward-Takahashi identities (or generalizations for non-Abelian gauge theories, called Slavnov-Taylor identities). These WTIs are also important for renormalizability: If a gauge theory is superficially renormalizable, i.e., contains only the allowed terms with not too many derivatives and/or fields, it is really renormalizable, although there are vertices that are not finite by power counting. E.g., in QED the four-photon vertex is of superficial degree of divergence 0 and thus naively logarithmically divergent. If it were really divergent, one would be in trouble since there is no gauge invariant local four-photon term for the Lagrangian which at the same time is renormalizable. Fortunately, the WTIs make the potentially divergent parts exactly 0 due to gauge symmetry, and QED is renormalizable. The same mechanism holds for spontaneously broken gauge symmetries ("Higgs mechanism") for both abelian and non-abelian gauge theories, leading to massive vector bosons without breaking gauge invariance.
In the abelian case, there's however another possibility, i.e., one can have a massive gauge boson, coupled to a conserved current (conserved because of a gauged global symmetry) without violating the gauge invariance due to the vector-boson mass term. This is known as the Stückelberg mechanism. That goes as follows: You start with the free Lagrangian for a massive vector field, i.e.,
\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{m^2}{2} A_{\mu} A^{\mu}.
This Lagrangian is not invariant under gauge transformations,
A_{\mu} \rightarrow A_{\mu}-\partial_{\mu} \chi
because of the mass term. This can be repaired by introducing a real scalar field \phi in the following way
\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{m}{2} A_{\mu} A^{\mu} + \frac{1}{2} (\partial_{\mu} \phi) (\partial^{\mu} \phi)-m \phi \partial_{\mu} A^{\mu}.
Then the Lagrangian is invariant under gauge transformations (up to a total divergence, i.e., the action is invariant)
A_{\mu} \rightarrow A_{\mu}-\partial_{\mu} \chi, \quad \phi \rightarrow \phi-m \chi.
Of course, this finally leads to a field theory of free massive vector bosons and a free massless scalar boson and is not very exciting.
However, of course, now you are free to introduce other particles. One application is the Kroll-Lee-Zumino model for \rho mesons, pions, and photons leading to the famous (renormalizable) vector-meson dominance model which gives good effective descriptions for the elastic pion-scattering phase shift in the 11-spin-isospin channel and the electromagnetic form factor of the pion. It's also easy to add \omega mesons.
Quantizing this kind of models with the Faddeev-Popov method allows a gauge fixing similar to the R_{\xi} gauges in electroweak theory, showing that indeed the interacting-particle spectrum consists of a massive vector meson and the matter fields. The field, \phi (in this context called Stückelberg ghost), decouples as the Faddeev-Popov ghosts do. The difference is that Faddev-Popov ghosts are scalar Grassmann fields while the Stückelberg ghost is a usual scalar field. Thus, the massive vector field together with the Stückelberg and Faddeev-Popov ghosts make 3 effective physical field degrees of freedom as it should be for a massive vector field. At the same time the model is gauge invariant very similar to QED and thus manifestly renormalizable in the R_{\xi} gauge.